Polyhedrality conjecture for cones of codimension-2 special cycles

Let $X$ be an orthogonal Shimura variety arising from an even unimodular lattice of signature $(b,2)$, where $b>2$. The cone of special cycles of codimension $2$ on $X$ is the cone generated by these codimension-$2$ cycles. Polyhedrality conjecture. The cone of special cycles of codimension $2$ on $X$ is polyhedral. The conjecture extends the known codimension-$1$ result for special divisors and is motivated by the arithmetic of Fourier coefficients of genus-$2$ Siegel modular forms; the paper provides supporting results and proves the first cases in low dimension.